In particular, what's the difference between uncertainty and measurement noise in engineering?
Noise is a concrete noun (for the want of the correct linguistic term). It is used to describe the background events that occur whether or not an experimental disturbance (producing specific signals of interest) is present.
Uncertainty is an abstract noun for the condition resulting from being unable to distinguish or separate the above signal from the “noise”.
Although the usage is technical, the grammatical difference between the two nouns should be evident.